Phase transitions in the Prisoner's Dilemma game on the Barab\'asi-Albert graph with participation cost

TL;DR

This study investigates how increasing social link maintenance costs influence cooperation levels in the Prisoner's Dilemma on scale-free networks, revealing abrupt phase transitions and metastable states through simulations and analytical methods.

Contribution

It demonstrates the critical role of participation costs in cooperation dynamics and characterizes the phase transition behavior on scale-free networks.

Findings

Cooperation frequency drops abruptly with increased link costs.

Near the critical point, the system exhibits bi-modal distributions and oscillations.

The critical region diminishes as population size grows, but metastability persists.

Abstract

We examine the impact of the maintenance cost of social links on cooperative behavior in the Prisoner's Dilemma game on the Barab\'asi-Albert scale-free network with a pairwise stochastic imitation. We show by means of Monte Carlo simulations and pair approximation that the cooperation frequency changes abruptly from an almost full cooperation to a much smaller value when we increase the cost of maintaining links. In the critical region, the stationary distribution is bi-modal and the system oscillates between two states: the state with almost full cooperation and one with coexisting strategies. We show that the critical region shrinks with the increasing size of the population. However, the expected time the system spends in a metastable state before switching to the other one does not change as a function of the system's size, which precludes the existence of two stationary states in the thermodynamic limit of the infinite population.